Question

In Problems 17-22, find the center and radius of the circle with the given equation. x^{2}+y^{2}-12 x+35=0

Short answer

The center is (6, 0) and the radius is 1.

Step-by-step solution

Identify the Standard Form of Circle Equation

The standard form of a circle equation is \((x-h)^2 + (y-k)^2 = r^2\), where \((h,k)\) is the center and \(r\) is the radius of the circle. Our goal is to rewrite the given equation in this form.

Reorganize the Equation

Given equation: \(x^2 + y^2 - 12x + 35 = 0\). Start by moving the constant term to the other side of the equation: \(x^2 + y^2 - 12x = -35\).

Complete the Square for the x-Term

To complete the square for \(x\)-terms, take the coefficient of \(x\), which is -12, divide by 2 to get -6, and square it to get 36. Add and subtract 36 inside the equation as follows: \((x^2 - 12x + 36) + y^2 = -35 + 36\).

Simplify the x-Term

The \(x\) terms are now in a perfect square form: \((x-6)^2 + y^2 = 1\). The \((x-6)^2\) represents the completed square for the \(x\)-terms.

Identify the Center and Radius

In the equation \((x-6)^2 + y^2 = 1\), the center \((h,k)\) is \((6,0)\) and the radius \(r\) is determined by taking the square root of 1, which is 1.

Key concepts

Completing the Square

Completing the square is a technique used in algebra to transform quadratic expressions into a perfect square trinomial. This method is useful for identifying properties such as the vertex in parabolas or rewriting equations in standard form like circle equations.

Here's how completing the square is applied in our example: For the given equation, we took the expression containing \(x\): \(x^2 - 12x\). The process involves these steps:

• Split the coefficient of \(x\), which is -12, into half: \(-12 / 2 = -6\).
• Square this result: \((-6)^2 = 36\).
• Add and subtract this square within the equation: \(x^2 - 12x + 36\).

This makes the term a perfect square trinomial, \((x-6)^2\). Adding \(36\) to both sides of the equation helps maintain equality, enabling us to clearly identify the circle's center and radius. Completing the square may initially sound complex, but mastering it simplifies many algebraic equations.

Circle Properties

A circle is defined as a set of points that are all equidistant from a central point, known as the center. The standard form of a circle's equation is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

In our example, by rewriting the equation \((x-6)^2 + y^2 = 1\), we identified the center at the point \((6, 0)\), and the radius, \(r\), is \(1\) as it's the square root of \(1\). Properties of a circle include:

• Diameter: Twice the radius \(2r\), in our example, it's \(2\).
• Circumference: The perimeter of the circle, calculated as \(2\pi r\).
• Area: The space within the circle, given by \(\pi r^2\).

Understanding these properties allows us to solve geometric problems and understand the spatial relationships in many real-world contexts.

Geometry

Geometry is the branch of mathematics dealing with shapes, sizes, and the properties of space. It encompasses various forms and dimensions, from lines to complex polygons, circles being one of the fundamental shapes studied for their unique properties.

Circles in geometry are known for their symmetry and uniform distance from the center in all directions. Understanding how to manipulate circle equations using geometric principles, like completing the square, reveals essential circle properties. This knowledge enhances spatial reasoning skills and applies to fields such as engineering and architecture.

Geometric concepts such as angles, lines, and curves converge in the study of circles. This intersection helps in uncovering relationships between different shapes and solving complex mathematical problems. With geometry as a base, one can explore and appreciate the intricate nature of mathematics.